Modified Riemann Liouville Fractional Derivative Research Articles

Homotopy Analysis Method using Jumarie’s Approach for Multi-Dimensional Nonlinear Schrödinger Equations

In this paper, we suggest a fractional functional for the Homotopy Analysis Method (HAM) to solve the nonlinear fractional order partial differential equations with fractional order initial conditions by using the modified Riemann–Liouville fractional derivative proposed by G. Jumarie. Graphs have been plotted for different values of α , which clearly reflect the reliability of proposed scheme.

Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method

The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel travelling wave solutions of the considered model are investigated by employing an effective analytic approach based on a complex fractional transformation and Jacobi elliptic functions. The extended Jacobi elliptic function method is a systematic tool for restoring many of the well-known results of complex fractional systems by identifying suitable options for arbitrary elliptic functions. To understand the physical characteristics of NFMT, the 3D graphical representations of the obtained propagation wave solutions for some free physical parameters are randomly drawn for a different order of the fractional derivatives. The results indicate that the proposed method is reliable, simple, and powerful enough to handle more complicated nonlinear fractional partial differential equations in quantum mechanics.

A new approach for the generalized fractional Casson fluid model with Newtonian heating described by the modified Riemann–Liouville fractional operator

This research is about the transfer of heat of a generalized fractional Casson fluid on an unsteady boundary layer that is passing through an infinite oscillating plate, in a vertical direction combined with the Newtonian heating. The results are obtained by using a Modified Riemann–Liouville fractional derivative. The present fluid model starts with the governing equations which are then converted to a system of partial differential equations (linear) by using some suitable nondimensional variables. Using the method of integral balance and the Laplace transform technique, an analytical solution is obtained. The velocity and temperature expressions are derived, and the effects of modeling parameters are shown in tables and graphs to validate the obtained theoretical results.

Investigation of adequate closed form travelling wave solution to the space-time fractional non-linear evolution equations

This work aims to construct exact solutions for the space-time fractional (2 + 1)- dimensional dispersive longwave (DLW) equation and approximate long water wave equation (ALW) utilizing the two-variable (G′/G,1/G)-expansion method and the modified Riemann–Liouville fractional derivative. The recommended equations play a significant role to describe the travel of the shallow water wave. The fractional complex transform is used to convert fractional differential equations into ordinary differential equations. Several wave solutions have been successfully achieved using the proposed approach and the symbolic computer Maple package. The Maple package program was used to set up and validate all of the computations in this investigation. By choosing particular values of the embedded parameters, we produce multiple periodic solutions, periodic wave solutions, single soliton solutions, kink wave solutions, and more forms of soliton solutions. The achieved solutions might be useful to comprehend nonlinear phenomena. It is worth noting that the implemented method for solving nonlinear fractional partial differential equations (NLFPDEs) is efficient, and simple to find further and new-fangled solutions in the arena of mathematical physics and coastal engineering.

Exact travelling wave solutions for nonlinear system of spatiotemporal fractional quantum mechanics equations

Schrödinger equation is an indispensable model for quantum mechanics, used for modelling several fascinating complex nonlinear physical systems, such as quantum condensates, nonlinear optics, hydrodynamics, shallow-water waves, and the harmonic oscillator. The objective of this paper is to investigate and study the exact travelling wave solutions of nonlinear triple fractional Schrödinger equations involving a modified Riemann–Liouville fractional derivative. Using the Riccati-Bernoulli Sub-ODE technique, the Bäcklund transformation is employed to handle the posed system. The traveling wave solutions methodology lies in converting the fractional Schrödinger equations into a nonlinear system of fractional ODEs. An infinite sequence of solutions to the fractional partial differential equations can be obtained directly through solving the resulting nonlinear fractional system. Some graphical representations of the obtained solutions after selecting suitable values for fractional values and parameters are illustrated to test accuracy and verify the power, and effectiveness of the proposed method.

Discrete soliton solutions of the fractional discrete coupled nonlinear Schrödinger equations: Three analytical approaches

The fractional discrete coupled nonlinear Schrödinger equations are solved on account of the modified Riemann–Liouville fractional derivative and Mittag–Leffler function. By using the fractional generalized Riccati method, generalized Mittag–Leffler function method and fractional generalized tanh–sech function method, some new analytical discrete solutions constructed by generalized trigonometric and hyperbolic functions are obtained, including discrete fractional bright soliton, dark soliton, combined soliton, and periodic solutions. In order to illustrate the effect of fractional order parameter on dynamics of fractional discrete soliton, some results in this study are illustrated graphically. Results indicate that although the combined wave is made up of singular coth function, it does not exhibit the singular property in the discrete case. Moreover, two kinds of scalar soliton solutions are found. These results could be of great significance to further study complex nonlinear discrete physical phenomena.

Propagation and interaction between special fractional soliton and soliton molecules in the inhomogeneous fiber

IntroductionFractional nonlinear models have been widely used in the research of nonlinear science. A fractional nonlinear Schrödinger equation with distributed coefficients is considered to describe the propagation of pi-second pulses in inhomogeneous fiber systems. However, soliton molecules based on the fractional nonlinear Schrödinger equation are hardly reported although many fractional soliton structures have been studied. ObjectivesThis paper discusses the propagation and interaction between special fractional soliton and soliton molecules based on analytical solutions of a fractional nonlinear Schrödinger equation. MethodsTwo analytical methods, including the variable-coefficient fractional mapping method and Hirota method with the modified Riemann–Liouville fractional derivative rule, are used to obtain analytical non-travelling wave solutions and multi-soliton approximate solutions. ResultsAnalytical non-travelling wave solutions and multi-soliton approximate solutions are derived. The form conditions of soliton molecules are given, and the dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed in the periodic inhomogeneous fiber and the exponential dispersion decreasing fiber. ConclusionAnalytical chirp-free and chirped non-traveling wave solutions and multi-soliton approximate solutions including soliton molecules are obtained. Based on these solutions, dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed. These theoretical studies are of great help to understand the propagation of optical pulses in fibers.

Fractional optical solitons with stochastic properties of a wick-type stochastic fractional NLSE driven by the Brownian motion

A new strategy is proposed to analytically construct stochastic fractional solutions of a wick-type SFNLSE driven from the Brownian motion by altogether using the Hermite transformation, modified Riemann–Liouville fractional derivative rule and fractional mapping method with white noise theory. These stochastic fractional solutions include fractional bright, dark solitons and trigonometric function solutions with or without chirped phase. By the dynamical analysis of stochastic solutions, it is found that the evolution of soliton shows randomness and random wave packets appear. Moreover, the randomness of integer order soliton evolution becomes more obvious compared with fractional solitons. They have discontinuous random phenomena with the evolution of time t, and some random independent wave packets appear. These results will play an important role in the related research on the physical application of the random model in the white noise environment.

The Riemann–Liouville fractional derivative for Ambartsumian equation

The Ambartsumian equation, based on the modified Riemann–Liouville fractional derivative, is analyzed in this paper. The solution is expressed as a power series of arbitrary powers and its convergence has been proven. In addition, we show that the present solution reduces to the results in the literature when the fractional derivative tends to 1. Moreover, the behavior of the obtained solution is discussed through figures.

Dynamical characteristic of analytical fractional solitons for the space-time fractional Fokas-Lenells equation

A new strategy exploiting together the modified Riemann–Liouville fractional derivative rule and two kinds of fractional dual-function methods with the Mittag–Leffler function is presented to solve fractional nonlinear models. As an example, the space-time fractional Fokas-Lenells equation is solved by this strategy, some new exact analytical solutions including bright soliton, dark soliton, combined soliton and periodic solutions are found. The comparison of two kinds of fractional dual-function methods is also presented. These solutions exist under a constraint among parameters of nonlinear dispersion, nonlinearity and self-steepening perturbation. In order to further study the optical soliton transport and better understand the physical phenomenon behind the model, dynamical characteristics of analytical fractional soliton solutions including some graphics and analysis is provided. The role of the fractional-order parameter is studied.

On the Dimensional Consistency Aware Fractional Domain Generalization of Simplest Chaotic Circuits

In this research, we generalize the simplest Chua’s chaotic circuit which is even more simpler than the four-element Chua’s circuit in terms of number of elements and the novel simplest chaotic circuit in the fractional domain by using the fractional circuit elements. Unlike the previous works, the time dimensional consistency aware generalization has been performed for the first time in this work. The dynamics of the generalized fractional nonlinear circuits have been analyzed by means of the fractional calculus based on the modified Riemann–Liouville fractional derivative where the Lyapunov exponents and dimensions have also been numerically calculated. We have found that including the dimensional consistency significantly alters the dynamic of the obtained fractional domain Chua’s circuit from that of the previous dimensional consistency ignored counterpart as different Lyapunov exponents and dimensions can be obtained. The conditions for both fractional domain circuits which cease to be chaotic have also been determined where such condition of Chua's circuit presented in this study is different from that of the previous work. This is because the time dimensionalconsistency has been included. The dynamical analyses of these circuits have also been performed where their conditions for being nonchaotic have been verified. Moreover, their emulators have also been realized.

Comments on “Different methods for [formula omitted]-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation” [Comput. Math. Appl. 71(6) (2016) 1259–1269

In this note, we show that the results in the above paper are inaccurate due to using the complex transform method for modified Riemann–Liouville fractional derivative which is an incorrect method.

Computational method for a fractional model of the helium burning network

Stellar cores may be considered as a nuclear reactor that play important role in injecting new synthesized elements in the interstellar medium. Helium burning is an important stage that contribute to the synthesis of key elements such as carbon, through the triple- α process, and oxygen. In the present paper, we introduce a computational method for the fractional model of the nuclear helium burning in stellar cores. The system of fractional differential equations is solved simultaneously using series expansion method. The calculations are performed in the sense of modified Riemann–Liouville fractional derivative. Analytic expressions are obtained for the abundance of each element as a function of time. Comparing the abundances calculated at the fractional parameter α=1, which represents the integer solution, with the numerical solution revealed a good agreement with maximum error ɛ=0.003. The product abundances are calculated at α=0.25,0.5,0.75 to declare the effects of changing the fractional parameters on the calculations.

A new analytical method for seeking traveling wave solutions of space–time fractional partial differential equations arising in mathematical physics

In this paper, based a new fractional sub-equation and the properties of the modified Riemann–Liouville fractional derivative, we propose a new analytical method named improved fractional (DαG/G) method for seeking traveling wave solutions of space–time fractional partial differential equations, which can be seen as the fractional version of the improved (G′/G) method, and is the improvement of the common fractional (DαG/G) method. Due to given traveling wave transformation, one certain space–time fractional partial differential equation can be converted into another fractional ordinary differential equation with respect to one new variable. For demonstrating the validity of this method, we apply it to seek exact traveling wave solutions for the (2+1)-dimensional space–time fractional Nizhnik–Novikov–Veselov System and the space–time fractional KP-BBM equation. With the aid of the mathematical software, some new exact traveling wave solutions including solitary wave solutions, periodic wave solutions and solutions with other forms for them are successfully found.

Generalized fractional supertrace identity for Hamiltonian structure of NLS–MKdV hierarchy with self-consistent sources

In the paper, based on the modified Riemann–Liouville fractional derivative and Tu scheme, the fractional super NLS–MKdV hierarchy is derived, especially the self-consistent sources term is considered. Meanwhile, the generalized fractional supertrace identity is proposed, which is a beneficial supplement to the existing literature on integrable system. As an application, the super Hamiltonian structure of fractional super NLS–MKdV hierarchy is obtained.

On the solutions of the nonlinear fractional differential equations via the modified trial equation method

In this study, the nonlinear fractional partial differential equations have been defined by the modified Riemann–Liouville fractional derivative. By using this fractional derivative and traveling wave transformation, the nonlinear fractional partial differential equations have been converted into nonlinear ordinary differential equations. The modified trial equation method is implemented to obtain exact solutions of the nonlinear fractional Klein–Gordon equation and fractional clannish random walker's parabolic equation. As a result, some exact solutions including single kink solution and periodic and rational function solutions of these equations have been successfully obtained. Copyright © 2015 John Wiley & Sons, Ltd.

Formulation and solution of space–time fractional Boussinesq equation

The fractional variational principles beside the semi-inverse technique are applied to derive the space–time fractional Boussinesq equation. The semi-inverse method is used to find the Lagrangian of the Boussinesq equation. The classical derivatives in the Lagrangian are replaced by the fractional derivatives. Then, the fractional variational principles are devoted to lead to the fractional Euler–Lagrange equation, which gives the fractional Boussinesq equation. The modified Riemann–Liouville fractional derivative is used to obtain the space–time fractional Boussinesq equation. The fractional sub-equation method is employed to solve the derived space-time fractional Boussinesq equation. The solutions are obtained in terms of fractional hyper-geometric functions, fractional triangle functions and a rational function. These solutions show that the fractional Boussinesq equation can describe periodic, soliton and explosive waves. This study indicates that the fractional order modulates the waves described by Boussinesq equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting waves are added by considering the fractional order derivatives beside the nonlinearity.

The [formula omitted]-expansion method using modified Riemann–Liouville derivative for some space-time fractional differential equations

In this paper, the fractional partial differential equations are defined by modified Riemann–Liouville fractional derivative. With the help of fractional derivative and traveling wave transformation, these equations can be converted into the nonlinear nonfractional ordinary differential equations. Then G′G-expansion method is applied to obtain exact solutions of the space-time fractional Burgers equation, the space-time fractional KdV-Burgers equation and the space-time fractional coupled Burgers’ equations. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. These results reveal that the proposed method is very effective and simple in performing a solution to the fractional partial differential equation.

FRACTIONAL TIME SCALE IN CALCIUM ION CHANNELS MODEL

We propose a fractional time scale to model the calcium ion channels model which is retarded with injection of calcium-chelator Ethylene Glycol Tetraacetic Acid (EGTA). By using this nonlinear time scale and Modified Riemann–Liouville fractional derivative, we convert the integer-order calcium ion channels model into fractional-order differential equations. We also analyze the range of order of fractional-order differential equations to ensure the equilibria of it are asymptotically stable. The simulation results are given to demonstrate the solutions of this model coincide with the real experiment data.

Some New Gronwall-Bellman-Type Inequalities Based on the Modified Riemann-Liouville Fractional Derivative

By use of the properties of the modified Riemann-Liouville fractional derivative, some new Gronwall-Bellman-type inequalities are researched. First, we derive some new explicit bounds for the unknown functions lying in these inequalities, which are of different forms from some existing bounds in the literature. Then, we apply the results established to research the boundedness, uniqueness, and continuous dependence on the initial value for the solution to a certain fractional differential equation.